VI CIDAMA 2014

One of the most important inequalities in modern convexity is Brunn-Minkowski inequality. Its functional analogue is Prekopa-Leindler inequality in $R^n$. We will see that, in a certain sense, we can consider it as a reverse of Hölder's inequality. We will study several consequences: the Brunn-Minkowski inequality for the Gaussian probability, the isoperimetric inequality in $R^n$ and in $S^{n-1}$.

The isoperimetric inequality in the Euclidean space has the counterpart of Sobolev inequality in the functional framework. We will consider another kind of isoperimetric inequalities, which will be Cheeger's type isoperimetric inequalities for log-concave probabilities in $R^n$ and their associated functional inequalities, which will be Poincaré type inequalities. Computing the corresponding spectral gap in Poincaré inequalities is the main problem in Kannan-Lovász-Simonivits conjecture, which will be explained.

In this course we will survey recent work on two weight norm inequalities for the fractional integral operator \[ I_\alpha f(x) = \int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n-\alpha}}\,dy, \qquad 0< \alpha < n, \] and its commutator with $BMO$ functions, \[ [I_\alpha, b ]f(x) = b(x)I_\alpha f(x) - I_\alpha (fb)(x). \]

We are interested in finding sufficient (and necessary and sufficient) conditions on pairs of weights $(u,\sigma)$ for the weak and strong-type inequalities \[ I_\alpha(\cdot \sigma) : L^p(\sigma) \rightarrow L^{q,\infty}(u), \quad I_\alpha(\cdot \sigma) : L^p(\sigma) \rightarrow L^{q}(u), \quad 1 < p \leq q < \infty. \]

Recently, using the machinery developed to prove the $A_2$ conjecture, there has been a great deal of progress in this area. We will first survey the history of this problem, starting with the work of Sawyer on testing conditions for pairs of weights $(u,\sigma)$: \[ \sup_Q \sigma(Q)^{-1/p}\left(\int_Q I_\alpha (\sigma\chi_Q)(x)^q u(x)\,dx \right)^{1/q} < \infty, \] where the supremum is taken over all cubes $Q$. We will then discuss the so-called $A_{p,q}$ bump conditions, \[ \sup_Q |Q|^{\alpha/n+1/q-1/p}\|u^{1/q}\|_{A,Q}\|\sigma^{1/p'}\|_{B,Q} < \infty, \] where $\|\cdot\|_{A,Q},\,\|\cdot\|_{B,Q}$ are normalized Orlicz norms. These conditions, which generalize the Muckenhoupt $A_p$ weights, were introduced by Pérez in the 1990's and are closely related to the recently disproved Muckenhoupt-Wheeden conjectures.

Throughout our talks we will discuss the parallels with recent work on singular integrals.

CANCELED

The topic is composition operators $f\longmapsto f\circ\varphi\,$, where the symbol $\varphi:{\mathbb D}\to{\mathbb D}$ is holomorphic. We shall give a (non-exhaustive) overview of -more or less recent- results when these operators are viewed on the classical Hardy spaces $H^p$. The story involves some classical tools of complex analysis, as Nevanlinna counting function and Carleson measures. We will illustrate this presentation with miscellaneous examples and questions. Concerning the most recent results, we shall pay attention to their possible membership to the class of absolutely summing operators.

Let $\mathbb{D}$ be the complex unit disc and let $A^p_\omega$ denote the Bergman space in the unit disc induced by a radial weight $\omega$ with the doubling property $\sup_{0\leq r<1}\frac{\int_{r}^1\omega(s)\,ds}{\int_{\frac{1+r}{2}}^1\omega(s)\,ds} < \infty$. To begin with, we shall present a decomposition norm theorem for $A^p_\omega$. This result will be used to obtain a description of the $L^p$-means and the $L^p_v$-behavior of $B^\omega _\zeta$, the reproducing kernels of $A^2_\omega$.

Later, we shall consider the Bergman projection from $L^2_\omega$ to $A^2_\omega$ \[ P_\omega (f)(z) = \int_{\mathbb D} f(\zeta)B^\omega_{\zeta}(z)\,\omega (\zeta)dA(\zeta), \] and study the two weight problem \[ \|P_\omega (f)\|_{L^p_v} \lesssim \|f\|_{L^p_v},\quad f\in L^p_v. \]

Joint works with O. Constantin and J. Rättyä.

1. O. Constantin and J. A. Peláez, Boundedness of the Bergman projection on $L^p$ spaces with exponential weights, submitted, available on http://arxiv.org/abs/1309.6071.
2. J.A. Peláez and J. Rättyä, Weighted Bergman spaces induced by rapidly increasing weights, Mem. Amer. Math. Soc. Vol. 227, n. 1066, (2014) http://arxiv.org/abs/1210.3311
3. J.A. Peláez and J. Rättyä, Generalized Hilbert operators on weighted Bergman spaces, Adv. Math. 240 (2013), 227-267.
4. J.A. Peláez and J. Rättyä, Two weight inequality for Bergman projection, preprint.

The meanings given to the terms "algebra" and "analysis", separately or jointly in expressions like "algebraic analysis" have changed over time, even simultaneously have been used with significant differentiating shades.

My presentation will focus on this issue during the period between the publication from 1751 of French Enlightenment's encyclopedia, L'Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers, and the first phase of the publication from 1899, of the German mathematician encyclopedia, Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, driven by F. Klein. The latter was translated into French with slight additions, under the direction of J. Molk, from 1904 until the Great War interrupted the process in 1915, when the period covered in my presentation ends.

Along the same, I will discuss some use cases of "algebra" and "analysis" with different meanings, and the different use of "algebraic analysis" in research and teaching.

Consider the Poisson equation $u_{tt} = \mathcal{L}u$ in ${{\mathbf{R}^{d+1}_{\scriptscriptstyle +}}}$, with $\mathcal{L}=-\Delta+|x|^2$ the Hermite operator. We look for very general conditions on the initial datum $f$, so that $u(t,x) = e^{-t\sqrt{-\mathcal{L}}}f(x)$ converges a.e. to $f(x)$. When $w(x)$ is a weight in $A_p$, this is classically obtained from the $L^p(w)$ boundedness of the associated maximal operators \[\mathcal{M}f(x)= \sup_{t > 0} |u(t,x)|.\] However, such convergence also holds with less restrictive conditions, such as boundedness from $L^p(w) \to L^p(v)$, for some other weight $v(x)$, of a local maximal operator $\mathcal{M}_{a}f= \sup_{0 < t < a} |u(t,x)|$ for some $a > 0$. This produces a larger class of weights than classical $A_p$ theory. In this work we solve this version of the 2-weight problem, and as a consequence characterize the weights $w(x)$ for which $u(t,x)\to f(x)$ a.e. for all $f\in L^p(w)$. The proof requires sharp estimates on the decay of Hermite-Poisson kernels, together with classical factorization techniques of Rubio de Francia. Similar results are also valid in the Ornstein-Uhlenbeck setting. This is part of the joint work with Hartzstein, Signes, Torrea, and Viviani.

An operator $T$ on a Fréchet space $X$ is called frequently hypercyclic if there is a vector $x\in X$ (also called frequently hypercyclic) such that, for any non-empty open set $U\subset X$, the set $\{n\geq 0 : T^nx\in U\}$ has positive lower density. We will discuss recent work on such operators. In particular, in joint work with A. Bonilla, we give a sufficient condition for the existence of a frequently hypercyclic subspace, that is, a closed infinite-dimensional subspace in which every non-zero vector is frequently hypercyclic. And Q. Menet has recently exhibited frequently hypercyclic operators that have a hypercyclic subspace but no frequently hypercyclic subspace.

We will discuss some recent results on abstract interpolation of linear as well as multilinear operators. In particular, we will present joint results with Loukas Grafakos on an abstract multilinear version of Stein's theorem for analytic families of multilinear operators defined on products of quasi-Banach spaces. We will show applications to the bilinear Hilbert transform and to the bilinear Bochner-Riesz operator.

In the early 90's, D. Sarason posed conjectures on the characterization of the boundedness of Toeplitz products on Hardy and Bergman spaces [3]. The Hardy space case attracted much attention because of its close relation to the famous two-weight problem for the Hilbert transform in Real Analysis, pointed out by Cruz-Uribe in [1]. Unfortunately, the Sarason conjecture for Toeplitz products on Hardy space was shown to be false by F. Nazarov around 2000 [2].

In this talk we will show that Sarason conjecture is also false in the Bergman space. Some aspects of the Bergman space setting are easier, because cancellation plays much less of a role in this setting, unfortunately the opposite happens when we look for a counterexample. We will also provide a characterization of the boundedness of Toeplitz products in the Bergman space in terms of testing conditions. This is a joint work with A. Aleman and S. Pott from Lund University.

1. David Cruz-Uribe, The invertibility of the product of unbounded Toeplitz operators, Int.Eq.Op.Th. 20 n.2 (1994), 231-237.
2. F. Nazarov, A counterexample to Sarason's conjecture, http://www.math.msu.edu/~fedja/prepr.html
3. D. Sarason, Products of Toeplitz Operators, Springer Lecture Notes in Mathematics, vol. 1573 (1994), 318-320

We shall discuss how to use semigroup theory in order to define the classical operators (Riesz transforms, square functions, Riesz potentials,...) associated to a general Laplacian. Several examples will be given. We shall focus in the special case of the discrete Laplacian in the integers. In the talk, we shall follow the path sketched by E. Stein in his celebrated monograph (cf. [1]). (See also [2]).

1. E. Stein, Topics in Harmonic Analysis, related to the Littlewood-Paley Theory, Annals of Mathematics Studies, Princeton University Press, Princeton, 1970.
2. O. Ciaurri, T.A. Gillespie, L. Roncal, J.L. Torrea, J.L. Varona; Harmonic Analysis associated with a discrete Laplacian, arXiv:1401.2091.

Bishop-Phelps Theorem states the denseness of the subset of norm attaining functionals in the (topological) dual of a Banach space. Bollobás proved a quantitative version of this result, which has been useful for numerical ranges of operators. Roughly speaking, Bollobás proved that each pair of elements $(x_0, x_{0}^*) $ in $S_X \times S_{X^*}$ such that $x_{0} ^* (x_0)$ is close to $1$ can be approximated by $(x,x^*)$ in $S_X\times S_{X^*}$ satisfying $x ^* (x)=1$. Recently the study of extensions of this result for operators was initiated. Since then some papers providing results for classical Banach spaces appeared. We will present recent results valid in case that the domain space of the operators is $C_0(L)$.

In this talk we will study the existence of a coincidence point for two mappings defined on a nonempty set and taking values on a Banach space using the fixed point theory for nonexpansive mappings. Using this type of results, we will obtain the existence of solutions for some classes of differential equations.

We say that a Banach space $X$ is nice whenever any extreme operator $T$ from a Banach space $Y$ to $X$ is a nice operator, that is, $T^*$, the adjoint of $T$, preserves extreme points. We get several necessary conditions for being nice. The main result is the characterization of nice finite-dimensional Banach spaces.

The Sobolev space $W^{1}L^{p}(I^{n}),$ $1\leq p\leq \infty,$ consists of all functions in $L^{p}(I^{n})$ whose first-order distributional derivatives also belong to $L^{p}(I^{n}).$ The classical Sobolev embedding theorem claims: \[ W^{1}L^{p}(I^{n})\hookrightarrow L^{pn/(n-p)}(I^{n}),\quad 1\leq p < n. \] Sobolev proved this embedding for $p > 1,$ but his method, based on integral representations, did not work when $p=1.$ That case was settled affirmatively by Gagliardo and Nirenberg, who first observed: \begin{equation}\tag{1} W^{1}L^{1}(I^{n})\hookrightarrow \mathcal{R}(L^{1},L^{\infty}), \end{equation} where $\mathcal{R}(L^{1},L^{\infty})$ is a mixed norm space, and then, using an iterated form of Hölder's inequality, completed the proof.

Our main goal in this work is to study the embedding (1) for more general rearrangement invariant (r.i.) spaces. In particular we concentrate on seeking the optimal domains and the optimal ranges for these embeddings between r.i. spaces and mixed norm spaces. As a consequence, we prove that the classical estimate for the standard Sobolev space $W^{1}L^{p}$ by Poornima and Peetre ($1 \leq p < n$), and by Hansson, Brézis, Wainger and Maz'ya ($p=n$) can be improved considering mixed norms as targets spaces.

This work is part of my PhD thesis, supervised by Javier Soria (University of Barcelona).

We characterize the class of weights related to the boundedness of maximal operators associated to Young functions of LlogL type in the context of variable Lebesgue spaces and we give sufficient conditions for more general Young functions. Fractional versions of these results are also obtained by means of a weighted Hedberg type inequality in the variable context. These results are new even in the classical Lebesgue spaces. We also deal with Wiener’s type inequalities for the mentioned operators in the spirit of the corresponding result proved in [CU-F] for the Hardy-Littlewood maximal operator. As applications of the strong type results for the maximal operators, we derive weighted boundedness properties for a large class of operators controlled by them, such as singular and fractional integrals with kernels satisfying certain Hörmander type condition and their commutators.

[CU-F] D. Cruz-Uribe and A. Fiorenza, $L\log L$ results for the maximal operator in variable $L^{p}$ spaces, Trans. Amer. Math. Soc. 361 (2009), no. 5, 2631-2647.

In this talk we present the study of the compactness of commutators of bilinear Calderón-Zygmund operators and their iterates with CMO symbols determining the suitable classes of multiple weights in which this property holds.

Joint work with Á. Bényi, K. Moen and R.H. Torres.

We will illustrate through examples the use of the reiteration theorems obtained in the joint papers with T. Signes [1,2,3]. These results are proved for interpolation methods defined by means of slowly varying functions and symmetric spaces. We will derive interpolation formulas for the couple $(L \log L , L_{exp})$ and other examples.

1. P. Fernández-Martínez, T. Signes, Real interpolation with symmetric spaces and slowly varying functions, Quart. J. Math., 63 No. 1, (2012), 133-164.
2. P. Fernández-Martínez, T. Signes, Limit cases of reiteration theorems, to appear in Math. Nachr.
3. P. Fernández-Martínez, T. Signes, A limit case of ultrasymmetric spaces, to appear in Arkiv der Matematik.

By means of rough convergence, we study two geometric properties in Banach spaces and relate them to Chebyshev centers and some well-known classical properties, such as Kalton's M property or Garkavi's uniform rotundity in every direction.

We present some examples of one-sided operators and focus our attention on the problem of characterizing the weak and strong type inequalities with weights for the one-sided Hardy-Littlewood maximal operator, in $\mathbb{R}$ and $\mathbb{R}^n$. In order to approach this problem we study several one-sided dyadic maximal operators.

A super-reflexive Banach space admits many uniformly convex equivalent norms. We prove that the set of all the moduli of convexity of this set of norms admits a supremum, in a quite natural function ordering. The classical result of Pisier about the uniformly convex renorming with modulus of power type follows easily from the properties of such a supremum.

In this talk we study ultrasymmetric sequence spaces in the case in which the fundamental function belongs to a limit class of concave functions. In the process we present a simple analytical description of these spaces and we establish new $J$-$K$ identities as well as a reiteration theorem for limit interpolation methods.

We also study ultrasymmetric approximation spaces and we give some applications to limit Lorentz-Zygmund operator ideals.

This is a joint work with Pedro Fernández-Martínez (Universidad de Murcia).

1. P. Fernández-Martínez, T. Signes,A limit case of ultrasymmetric sequence spaces, preprint.
2. E. Pustylnik, Ultrasymmetric sequence spaces in approximation theory, Collect. Math. 57 (3) (2006), 257-277.